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📰 Solution: By De Moivre's Theorem, $z^6 = \cos(6\theta) + i\sin(6\theta) = -1 + 0i$. This implies $6\theta = \pi + 2\pi k$ for integer $k$. Solving for $\theta$ gives $\theta = \frac{\pi}{6} + \frac{\pi k}{3}$. The principal solution in $[0, 2\pi)$ is $\theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6}$. The smallest positive $\theta$ is $\boxed{\dfrac{\pi}{6}}$. 📰 Question: A tech entrepreneur develops a sensor system where two unit vectors $\mathbf{p}$ and $\mathbf{q}$ form a plane. If $\mathbf{p} \times \mathbf{q} = \begin{pmatrix} 0 \\ 0 \\ \frac{1}{2} \end{pmatrix}$, find the angle between $\mathbf{p}$ and $\mathbf{q}$. 📰 Solution: The magnitude of the cross product is $\|\mathbf{p} \times \mathbf{q}\| = \|\mathbf{p}\| \|\mathbf{q}\| \sin\theta$. Since $\mathbf{p}$ and $\mathbf{q}$ are unit vectors, $\|\mathbf{p} \times \mathbf{q}\| = \sin\theta = \frac{1}{2}$. Thus, $\theta = \arcsin\left(\frac{1}{2}\right)$, which gives $\theta = \frac{\pi}{6}$ or $\frac{5\pi}{6}$. However, the angle between vectors is taken as the acute angle, so $\theta = \boxed{\dfrac{\pi}{6}}$. 📰 Getvelocity 4531661 📰 Jimmy Kimmel Comments On Charlie Kirk 4454331 📰 Kane And Lynchs Hidden Secrets Revealedyou Wont Believe What Fans Missed 7456539 📰 David Cages Secret Mastermind Move You Wont Guess What Defined His Creative Genius 6858065 📰 Cowboy Hat Drawing 3834255 📰 Whats Really Inside Avocado Oil That Olive Oil Swears Cant Match 3171007 📰 Abington Sr High 4923780 📰 Flights From Nashville To Las Vegas 7039294 📰 You Wont Believe Whats Hidden In Portlands Craigslist 2165893 📰 What Bashas Weekly Secret Keeps Everyone Talking Across The Community 9955107 📰 Connections Hint Oct 21 5841965 📰 Top Restaurant Games Proven To Keep You Entertained For Hourstry Them Now 5123732 📰 Regn Stock Price 5754092 📰 Black Glasses You Cant Resistdiscover The Accessory Trending This Season 4495971 📰 Why Every Modern Living Room Needs A Coffee Table In Travertinea Quick Trend Fix 1539474